Question: $ A = \left[\begin{array}{rrr}0 & -2 & -2 \\ 3 & -2 & 0\end{array}\right]$ $ C = \left[\begin{array}{rr}-2 & 4 \\ 4 & 0 \\ 2 & -1\end{array}\right]$ Is $ A C$ defined?
In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ A$ , have? How many rows does the second matrix, $ C$ , have? Since $ A$ has the same number of columns (3) as $ C$ has rows (3), $ A C$ is defined.